Abstract
This paper is concerned with a convex polynomial optimization problem in the face of data uncertainty both in the objective function and the constraints. Using the minimax robust optimization approach, we formulate the robust counterpart of the uncertain convex polynomial optimization problem. First, we present dual characterizations of positivity and nonnegativity of the objective function of a convex polynomial optimization problem with data uncertainty. Next, based on the dual characterizations, we derive optimality conditions for the robust convex polynomial optimization problem. Then, we consider a semidefinite programming (SDP) relaxation of the robust convex polynomial optimization problem, and show the convergence of an SDP hierarchy by means of optimality conditions derived for an equivalent reformulation of the robust problem. Finally, under an additional assumption that the Hessian matrix of the Lagrange function of the uncertain convex polynomial optimization problem is positive definite at some saddle point, we prove the finite convergence of an SDP hierarchy of the robust convex polynomial optimization problem.
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The authors would like to thank two anonymous referees and the editors for their helpful comments and suggestions which have led to the improvement of the earlier version of this paper.
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Huang, L., Liu, D. & Fang, Y. Convergence of an SDP hierarchy and optimality of robust convex polynomial optimization problems. Ann Oper Res 320, 33–59 (2023). https://doi.org/10.1007/s10479-022-05103-6
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DOI: https://doi.org/10.1007/s10479-022-05103-6